The full posterior distribution corresponding to the parameters of the model in Equation (REF ) with the prior structure considered in Equation (REF ) is given in Section S2.1. Due to the multiple-multivariate regression setup, we have \(g \times p^{(t)}\)  coefficients (\(\beta _{kj}\) ) and/or \(|M| \times g\)  posterior inclusion probabilities to sequentially estimate for each model at layer \(t\) . A full MCMC-based estimation can be employed, but the computation time would be additive with each additional spherical layer included, as the estimation is required to be sequential based on the model formulation. As a faster alternative to MCMC, we consider the EM-based strategy to variable selection as proposed by [1]}. This EM-based estimation approach is a deterministic alternative to commonly used stochastic search algorithms, which exploits the prior formulation to rapidly find posterior modes. We emphasize that the EM-based estimation presented in this work is an adaptation of [1]} to the multiple-multivariate regression setting. We discuss sequential estimation using EM and describe the model selection procedure below.
